Setup and Rule Diagrams

[Jon Denning]

Setup and Rule Diagram Explanation

This is a Defined-Fixed, Balanced, Grouping Game.


This explanation is still in progress. Please post any questions below!




Main diagram:

g4.png

[lathlee]

What game is this ? Grouping game ? grouping with pattern element ?

[Jon Denning]

Just Grouping! The oddness here—in my opinion, anyway—comes from the numerical idea that the "Both" pair includes the 5 and the 10, as well. So saying a corporation offers a 5-year bond allows it to be exclusively 5, or possibly Both.

That's why I've shown two of each type in the diagram, to make it clear there aren't really four 5-year spots, but really two that are strictly 5, and two more that are Both. Ditto for 10.

It's also why in writing out the conditional rules (like rule #2) I included "Both" when the language didn't make it exclusively 5 or 10. In fact, it's really only when you're told a certain type of bond is NOT offered that you can rule out the "Both" possibility, which you see now and then in the contrapositives I've listed

[CPA2lawschool]

Main diagram:

g4.png

Hi Jon,

Thanks for providing this. Would you mind to share your thoughts on what led you to use this setup (i.e. 5, 5, 10, 10, B, B) as compared to other methods (I opted for two horizontal rows (5 and 10) of four with blocks around the first and second spaces to denote both during my attempt).

The two keys to the setup of this game appear to be (i) approaching with a condensed version of "both" and (ii) an understanding that "both" is an option for each variable (V and R most notably) unless explicitly stated that it cannot be both.

Thanks in advance!

[Malila Robinson]

Hi CPA2lawschool,

If I understand the way you set the game up, I don't think there would be a problem with your set-up. Some games can be set-up in multiple ways ***that should not be a hindrance to your time and inferences***.

For grouping games without a linear element it is still ok to list them out in a single line as long as the spaces are labeled, just as you would label a 1-8 linear game. For this game that would lead to 2 spaces each for 5, 10 and B (both). This set-up has a possibly confusing element in it because the test taker would need to understand that the 2 spaces in B would really be 2 variables that are listed twice for a total of 4. Without that understanding, the chart set-up in the first post may look like it is only listing 6 variables. But for people with that understanding, a 6 variable chart may look like it is easier to work with than an 8 variable chart.
Hope that helps!
-Malila

[Ari]

Hello,

I could really use some help understanding the contrapositives of the conditionals for rules two and three. I feel like I am overlooking something obvious, but I have tried to step away and come back to this problem and I cannot see how we can infer the Rb + Gb in rule two and the H10 + Sb in rule three. Can someone please help me out? Thank you in advance!

Ari

[Jeremy Press]

Hi Ari,

Those are quite tricky, so you're definitely getting to the heart of the hardest part of this game!

Let's say, for the second rule, that S does not offer both types of bonds. That means that V cannot offer 5-year bonds (and cannot offer both), and must offer 10-year bonds alone. That fills out the diagram (according to Jon's original sketch) like so:

Screen Shot 2020-08-28 at 3.45.29 PM.png

Since S and V now cannot offer both types, and H/L can never offer both types, there are only two remaining companies that can offer both types: R and G.


Moving on to the third rule, the contrapositive begins from R offering 5-year bonds (but not 10-year and not both), which means L must not offer 10-year bonds, and must therefore offer 5-year bonds. Since L and H cannot offer the same type, H now has to offer 10-year bonds, diagrammed per below:

Screen Shot 2020-08-28 at 3.45.36 PM.png

The S inference is a little head-spinning, admittedly. But, think about what would happen if S offered only 10-year bonds. That would mean V would have to offer both types, which would trigger the second rule (V offering 5-year) and force S to offer both types. So, S has to offer both types!

Let us know if that clears it up!

[CodeyD29]

Can someone please tell me why you chose the bonds as the base opposed to the corporations? I seem to make this mistake often and it's bugging me.

[Adam Tyson]

My reason for using the bonds as the base in this game is that they have established numeric values, CodeyD29. Using the base from the explanation here, you can see that there are exactly two 5s, two 10s, and two "boths'. If we use the corporations as the base we can never be sure how many bonds each corporation offers - it could be one or it could be two. Having a fixed numeric base can be very useful!

An alternative setup to this one is to set up two groups, a group for 5 year bonds and one for 10 year bonds. Each group would have exactly four members in it - again, certainty in the numbers makes for a good base. Then you can start figuring out which corporations are in just one group or just the other group and which are in both groups. I also like doing this one as a template game based on which two corporations offer both types. If, for example G and R are the ones that go twice, then S does not go in both groups, and that means V cannot be in the 5 group and must be only in the 10 group.

Think about the numbers! They will often guide you to a better base in your setup.

[CodeyD29]

Thank you very much Adam, I really appreciate it.